Decision Support
A clustering procedure for reducing the number of representative solutions in the Pareto Front of multiobjective optimization problems

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Abstract

In many multiobjective optimization problems, the Pareto Fronts and Sets contain a large number of solutions and this makes it difficult for the decision maker to identify the preferred ones. A possible way to alleviate this difficulty is to present to the decision maker a subset of a small number of solutions representatives of the Pareto Front characteristics.

In this paper, a two-steps procedure is presented, aimed at identifying a limited number of representative solutions to be presented to the decision maker. Pareto Front solutions are first clustered into “families”, which are then synthetically represented by a “head-of-the-family” solution. Level Diagrams are then used to represent, analyse and interpret the Pareto Front reduced to its head-of-the-family solutions. The procedure is applied to a reliability allocation case study of literature, in decision-making contexts both without or with explicit preferences by the decision maker on the objectives to be optimized.

Introduction

Multiobjective decision-making can be aided by methods which provide flexible ways of handling multiple objectives and decision maker (DM) preferences, rather than by quasi-prescriptive methods based on the aggregation of the multiple objectives into a single one. In this respect, the importance of determining and including DM preferences for the practice of multiobjective decision making and optimization is inarguable.

The particular context of reference for the present work are practical decision-making situations concerning high-consequence technologies, e.g. nuclear, oil and gas, transport etc. The starting point is the acknowledgment that technical analyses provide useful decision support in the sense that their outcomes inform the decision makers insofar as the technical side of the problem is relevant for the decision.

It is further understood that the actual decision outcome for a critical situation involving a potential for large consequences typically derives from a thorough process which combines (i) an analytic evaluation of the situation (i.e., the technical assessment) by rigorous, replicable methods evaluated under agreed protocols of an expert community and peer-reviewed to verify the assumptions underpinning the analysis, and (ii) a deliberative group exercise in which all involved stakeholders and decision makers collectively consider the decision issues, look into the arguments for their support, scrutinize the outcomes of the technical analysis and introduce all other values (e.g. social and political) not explicitly included in the technical analysis. This way of proceeding allows keeping the technical analysis manageable by complementation with deliberation for ensuring coverage of the non-modelled issues. In this way, the analytic evaluation (i.e., the technical assessment) supports the deliberation by providing numerical outputs of the relevant parameters, possibly to be compared with predefined numerical safety criteria for further guidance to the decision, and also all the argumentations behind the analysis itself, including the assumptions, hypotheses, parameters and their uncertainties.

The ultimate concern of the DM is to confidently fulfil his or her conflicting objectives, while satisfying the constraints posed by the problem itself.

In practice, the technical assessment amounts to the solution of a multiobjective optimization problem in terms of a discrete approximation of the Pareto Front and corresponding Pareto Set of solutions. The ultimate purpose of the technical assessment is to provide the DM with a clearly informed picture of the problem upon which he or she can confidently reason and deliberate. On the basis of the information provided by the technical assessment, the DM is requested to select one or more feasible solutions according to criteria which depend on the decision situation. In the literature, it is well acknowledged that presenting the DM with too many alternatives increases the burden of his or her decision-making task.

Different approaches exist for introducing DM preferences in the optimization process; a common classification is based on when the DM is consulted: a priori, a posteriori, or interactively during the search. A priori methods use DM preferences to bias the search of optimal solutions towards a preferred region, for example by changing the definition of dominance (Molina et al., 2009, Zio et al., 2009), by weighting differently the objectives (Yang, 1996), by assigning reference values (goals) and priority levels to the objectives (Yang, 2000), by assuming a utility function describing the DM behaviour and interest in the alternative solutions (Malakooti, 1988). Interactive methods require the direct intervention of the DM in the optimal solution search, for example simply to stop an iterative trial-and-error search when satisfactory results are reached (Katagiri et al., 2008) or more effectively to drive the optimization by ranking and eliminating alternatives based on indicated preference strengths (Roy, 1968a, Roy, 1968b, Roy, 1974, Roy and Bouyssou, 1986; Malakooti, 1988, De Boer et al., 1998) or by bounding DM utility functions by elicited preference information (Cho and Kim, 1997, Rios Insua and Martin, 1994), while accounting for the fact that the consequences of the alternative solutions may not be completely known, the problem definition may not be exact and the DM preferences may be only partially known and even partially inconsistent. A posteriori methods, on the other hand, apply DM preferences only after the optimal solutions of the Pareto Front are found.

The selection task by the DM can be difficult when the Pareto Front contains a large number of solutions. To make the task feasible, only a small number of solutions representatives of the Pareto Front should be offered for selection to the DM.

At the same time, it is important for the DM to be able to analyse the Pareto Front to evaluate the quality and adequateness of the results. For this reason, the number of solutions that the Pareto Front is reduced to has to be determined carefully: a number of solutions too small might not be sufficiently representative and informative, whereas a number too large might still be intractable for decision making purposes.

In this work, an a posteriori procedure is proposed for reducing the set of Pareto solutions on a Pareto Front to a small number of representative ones. Two situations are considered, depending on the presence or absence of explicit preferences of the DM on the objectives of the optimal decision. The procedure is made of two main steps. First, the set of optimal solutions constituting the Pareto Front and Set is partitioned in a number of clusters (here also called “families”) of solutions sharing common features. The clustering is performed by considering the distance between solutions in the objective values space; to this purpose, different clustering algorithms are available, e.g., the k-means (Bandyopadhyay and Maulik, 2002), the c-means (Bezdek, 1974) and subtractive clustering (Chiu, 1994); in this paper, the latter algorithm is used for the technical reasons explained in Section 3. The second step consists in selecting for each family (or cluster) the representative solution (the “head of the family”): depending on the decision situations (i.e., the presence or the absence of preferences on the objectives), different selection criteria may apply to provide the DM with the best solutions according to his or her requirements.

The outcome of the procedure is a Pareto Front reduced to a number of representative solutions, thanks to the clustering technique, which are ranked either by a specific norm or by DM explicit preferences, if available. In both cases, one may actually proceed to identify the best solution, i.e. the highest ranked according to the specified norm or DM preferences. However, in various practical decision making situations it is important that the DM has a picture of the spectrum of solutions available on the Pareto Front, for a solid support to his or her decision or for considering alternative compromises and preferences on the objectives in light of the Pareto Front obtained. Within a deliberative process of decision making, supported by the quantitative analysis performed, knowledge of the information contained in the Pareto Front and Set allows critically discussing, questioning, revising the preferences assumed and defending, supporting the choices made, possibly even non-optimal for the given situation a posteriori of consideration of additional aspects of the situation involved in the decision.

The originality of the work mainly lies in the following aspects:

  • the proposal of using the ideal solution (i.e., the solution which is optimal with respect to all the objectives simultaneously) for identifying the representatives of the clusters in decision situations in which the DM does not express preferences on the objectives of the optimization; such proposal stands on a definition of a 1-norm to measure the distance of the solutions from the ideal one and provides an algorithmic generalization of the empirical method introduced by the authors in (Zio and Bazzo, 2010a);

  • the effective integration of a fuzzy scoring procedure introduced by the authors in (Zio and Bazzo, 2010b) for ranking the cluster representative solutions in situations in which the DM expresses preferences on the objectives of the optimization;

  • the analysis of the clustered Pareto Front and Set within a Level Diagrams representation.

These developments render the proposed procedure of general applicability with respect to both the decision situations and the Pareto Front characteristics.

The procedure is applied to a reference case study regarding a redundancy allocation problem of literature with three objectives: system availability to be maximized, system cost and weight to be minimized (Taboada and Coit, 2007).

Level Diagrams (Blasco et al., 2008) are used to graphically represent, analyse and interpret the Pareto Front and Set considered in the analyses.

The remainder of the paper is organized as follows: Section 2 presents upfront the case study; Section 3 contains the analysis of the clustering algorithm, the methods considered for selecting the representative heads of the families and the results of their application to the case study; Section 4 provides a critical discussion of the results; Section 5 gives the conclusions that can be drawn from the findings of the work.

Section snippets

Case study: redundancy allocation in a multistate system

The case study is taken from (Taboada and Coit, 2007); it is a system reliability design problem with three conflicting objectives: system availability to be maximized; system cost and weight to be minimized. The system is made of u = 5 units (subsystems) connected in series; each unit can be provided with redundancy by selecting components from mp types available in the market, p = 1,  , 5. Each component is binary, i.e., at any time it can be in only two states: functioning at nominal capacity or

Subtractive clustering algorithm

The first step of the procedure for reducing the number of solutions to present to the DM is to group the solutions of the Pareto Front in a number K of families of solutions sharing similar characteristics. In this work, subtractive clustering (Chiu, 1994) is used to identify the families of similar solutions (clusters) Fj, j = 1,  , K, in the objective function space. The clustering is performed on the basis of the distance between solutions, i.e., in this case, between objective function values.

Discussion

The proposed procedure for reducing the solutions of the Pareto Front and Set for presentation to the DM is summarized in Fig. 9.

The combination of the clustering algorithm and the selection of the minimum 1-norm solutions as representative of the clustered families is a sound generalization of the empirical method introduced by the authors in (Zio and Bazzo, 2010a), where families of solutions were identified by looking for “vertical alignments” in the 1-norm Level Diagram of the availability

Conclusions

Multiobjective decision-making is the process of choosing a possible course of action among the alternative solutions available, which are judged preferentially by one or more DMs with respect to several conflicting criteria. In fulfilling the conflicting goals, the DMs must account also for the constraints imposed by the system itself.

The issue can be formulated in terms of a multiobjective optimization problem whose solving produces a Pareto Set of non-dominated solutions among which the DMs

Acknowledgments

The authors are thankful to Professor David Coit of Rutgers University for providing the Pareto Front and Set data of the case study, and to the three anonymous reviewers for providing in-depth comments which have stimulated a thorough revision of the paper, for its improvement.

Enrico Zio (BS in Nuclear Engng., Politecnico di Milano, 1991; MSc in Mechanical Engng., UCLA, 1995; PhD, in Nuclear Engng., Politecnico di Milano, 1995; PhD, in Nuclear Engng., MIT, 1998) is Director of the Chair in Complex Systems and the Energetic Challenge of Ecole Centrale Paris and Supelec, Director of the Graduate School of the Politecnico di Milano, full professor of Computational Methods for Safety and Risk Analysis, adjunct professor in Risk Analysis at the University of Stavanger,

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    Enrico Zio (BS in Nuclear Engng., Politecnico di Milano, 1991; MSc in Mechanical Engng., UCLA, 1995; PhD, in Nuclear Engng., Politecnico di Milano, 1995; PhD, in Nuclear Engng., MIT, 1998) is Director of the Chair in Complex Systems and the Energetic Challenge of Ecole Centrale Paris and Supelec, Director of the Graduate School of the Politecnico di Milano, full professor of Computational Methods for Safety and Risk Analysis, adjunct professor in Risk Analysis at the University of Stavanger, Norway, and invited lecturer and committee member at various Master and PhD Programs in Italy and abroad.

    He has served as Vice-Chairman of the European Safety and Reliability Association, ESRA (2000–2005) and as Editor-in-Chief of the International journal Risk, Decision and Policy (2003–2004). He is currently the Chairman of the Italian Chapter of the IEEE Reliability Society (2001).

    He is member of the editorial board of the International Scientific Journals Reliability Engineering and System Safety, Journal of Risk and Reliability, Journal of Science and Technology of Nuclear Installations, plus a number of others in the nuclear energy field.

    He has functioned as Scientific Chairman of three International Conferences and as Associate General Chairman of two others, all in the field of Safety and Reliability.

    His research topics are: analysis of the reliability, safety and security of complex systems under stationary and dynamic operation, particularly by Monte Carlo simulation methods; development of soft computing techniques (neural networks, fuzzy logic, genetic algorithms) for safety, reliability and maintenance applications, system monitoring, fault diagnosis and prognosis, and optimal design.

    He is co-author of three international books and more than 150 papers on international journals, and serves as referee of more than 20 international journals.

    Roberta Bazzo (BS in Energy Engineering, Politecnico di Milano, 2007, MS in Nuclear Engineering, Politecnico di Milano, 2009). Her interests include Pareto Front and Set analyses for decision making in RAMS and banking applications.

    This work has been partially funded by the Foundation pour une Culture de Securité Industrielle of Toulouse, France, under the research contract AO2009-04.

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